ProofUsingSAS and HL
LearningObjectives
- Understandand applythe SAS CongruencePostulate.
- Identifythe distinctcharacteristicsand propertiesof right triangles.
- Understandand applythe HL CongruenceTheorem.
- Understandthat SSA doesnot necessarilyprovetrianglesare congruent.
Introduction
You havealreadyseenthreedifferentwaysto provethat two trianglesare congruent(withoutmeasuring
six anglesand six sides).Sincetrianglecongruenceplayssuchan importantrole in geometry, it is important
to knowall of the differenttheoremsand postulatesthat can provecongruence,and it is importantto know
whichcombinationsof sidesand anglesdonotprovecongruence.
SAS Congruence
By now, you are very familiarwith postulatesand theoremsusingthe letters and to representtriangle
sidesand angles.One moreway to showtwo trianglesare congruentis by theSAS CongruencePostulate.
SAS TriangleCongruencePostulate:If two sidesand the includedanglein one triangle
are congruentto two sidesand the includedanglein anothertriangle,then the two triangles
are congruent.Like ASA and AAS congruence,the orderof the lettersis very significant.You musthavethe anglesbetween
the two sidesfor the SAS postulateto be valid.
Onceagainyou can test this postulateusingphysicalmodels(suchas piecesof uncookedspaghetti)for
the sidesof a triangle.You’ll find that if you maketwo pairsof congruentsides,and lay themout with the
sameincludedanglethen the third side will be determined.